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MATH408: Probability &amp; Statistics Summer 1999 WEEK 5

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MATH408: Probability &amp; Statistics Summer 1999 WEEK 5. Dr. Srinivas R. Chakravarthy Professor of Mathematics and Statistics Kettering University (GMI Engineering &amp; Management Institute) Flint, MI 48504-4898 Phone: 810.762.7906 Email: schakrav@kettering.edu

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MATH408: Probability & StatisticsSummer 1999WEEK 5

Dr. Srinivas R. Chakravarthy

Professor of Mathematics and Statistics

Kettering University

(GMI Engineering & Management Institute)

Flint, MI 48504-4898

Phone: 810.762.7906

Email: schakrav@kettering.edu

Homepage: www.kettering.edu/~schakrav

Joint PDF
• So far we saw one random variable at a time. However, in practice, we often see situations where more than one variable at a time need to be studied.
• For example, tensile strength (X) and diameter(Y) of a beam are of interest.
• Diameter (X) and thickness(Y) of an injection-molded disk are of interest.
Joint PDF (Cont’d)X and Y are continuous
• f(x,y) dx dy = P( x < X < x+dx, y < Y < y+dy) is the probability that the random variables X will take values in (x, x+dx) and Y will take values in (y,y+dy).
• f(x,y)  0 for all x and y and
Independence

We say that two random variables X and Y are independent if and only if

P(XA, YB) = P(XA)P(YB) for all A and B.

Groundwork for Inferential Statistics
• Recall that, our primary concern is to make inference about the population under study.
• Since we cannot study the entire population we rely on a subset of the population, called sample, to make inference.
• We saw how to take samples.
• Having taken the sample, how do we make inference on the population?

measurement in Example 3-33.

8 pull-off force measurements in Example 3-33.

function of the sample variance of 8 pull-off force

measurements in Example 3-33.

Central Limit Theorem
• One of the most celebrated results in Probability and Statistics
• History of CLT is fascinating and should read “The Life and Times of the Central Limit Theorem” by William J. Adams
• Has found applications in many areas of science and engineering.
CLT (cont’d)
• A great many random phenomena that arise in physical situations result from the combined actions of many individual ones.
• Shot noise from electrons; holes in a vacuum tube or transistor; atmospheric noise, turbulence in a medium, thermal agitation of electrons in a conductor, ocean waves, fluctuations in stock market, etc.
CLT (cont’d)
• Historically, the CLT was born out of the investigations of the theory of errors involved in measurements, mainly in astronomy.
• Abraham de Moivre (1667-1754) obtained the first version.
• Gauss, in the context of fitting curves, developed the method of Least Squares, which lead to normal distribution.
HOMEWORK PROBLEMS

Sections 3.11 through 3.12

109,111, 114-116-119, 121-123, 129-130

Tests of Hypotheses

• Two types of hypotheses: Null (H0)and alternative (H1)
Basic Ideas in Tests of Hypotheses
• Set up H0 and H1. For a one-sided case, make sure these are set correctly. Usually these are done such that type 1 error becomes “costly” error.
• Choose appropriate test statistic. This is usually based on the UMV estimator of the parameter under study.
• Set up the decision rule if  = P(type 1 error) is specified. If not, report a p-value.
• Choose a random sample and make the decision.
Setting up Hoand H1
• Suppose that the manufacturer of airbags for automobiles claims that the mean time to inflate airbag is no more than 0.1 second.
• Suppose that the “costly error” is to conclude erroneously that the mean time is < 0.1.
• How do we set up the hypotheses?
Test on µ using normal
• Sample size is large
• Sample size is small, population is approximately normal with known .

DNR Region

µ

CP_1

CP_2

Example (page 142)
• µ = Mean propellant burning rate (in cm/s).
• H0:µ = 50 vs H1:µ  50.
• Two-sided hypotheses.
• A sample of n=10 observations is used to test the hypotheses.
• Suppose that we are given the decision rule.
• Question 1: Compute P(type 1 error)
• Question 2: Compute P(type 2 error when µ =52.
Confidence Interval
• Recall point estimate for the parameter under study.
• For example, suppose that µ= mean tensile strength of a piece of wire.
• If a random sample of size 36 yielded a mean of 242.4psi.
• Can we attach any confidence to this value?
• Answer: No! What do we do?
Confidence Interval (cont’d)
• Given a parameter, say, , let denote its UMV estimator.
• Given , 100(1-  )% CI for is constructed using the sampling (probability) distribution of as follows.
• Find L and U such that P(L < < U) = 1- .
• Note that L and U are functions of .